A ONE-DIMENSIONAL MODEL WITH PHASE-TRANSITION

Two repellent particles are bound to occupy two among the k(n) + 1 adj acent sites 0 = x0(n) < x1(n) < ... < x(kn)(n) = 1, say x(q)(n), x(q+1 )(n). Define the Hamiltonian H(q)(n) = -ln(x(q+1)(n) - x(q)(n)) and th e partition function Z(beta, n) = SIGMA/0 less-than-or-equal-to q < k( n) exp{- betaH(q)(n)}. We discuss the behaviour of the function [GRAPH ICS] closely related to the free energy. We prove that the smallest re al zero of F(beta) is equal to the fractal dimension of the system and that this number, when less than one, is a critical value where F is not analytic.